Timothy stood proudly at the front of the room admiring the look of everything. Timothy had completely rearranged his classroom. He had grown bored with the tables in rows all facing toward the blackboard at the front of the room. He believed that it didn't make for an intriguing and free learning environment. It was far too uniform and far too stuffy. And Timothy hated uniformity. This year, he decided to approach things differently. The tables were arranged in triangular formations in the room. Comprising of three tables, each able to sit two students, the triangular shaped pods were able to seat six students total. The intention with the formation is to allow the students to feel as they were a community of minds rather than just students. It also opened the room up considerably allowing for free movement without hinderance. Timothy, as he usually did, had the shutters on the windows open to allow the natural light in to warm the room. Sitting on every table was a little glass bowl of sweets for the student's taking. It was comfortable and very unlike a classroom.

Timothy opened all of the shutters and windows in the room, inviting in natural light and the cooling September breeze to carry through the room. Timothy had learned a long time ago that allowing an open space and feeling in the room was incredibly stimulating to the mind. On the blackboard at the front of the room was Critical Thinking Arithmetic Principals. This was the first year that Hogwarts had made the distinction between intermediate and advanced learning in this subject and Timothy was thrilled with the opportunity to tackle some more challenging aspects in this field. He had been reluctant to raise the difficulty in previous years as he knew many students in the room weren't ready for that. But with these new advanced classes, he knew he could stretch his legs a bit more.

As the students filed in, Timothy smiled widely. He welcomed each student as they entered. When class was ready to begin, he closed the door. "Hello, hello! Thank you for your punctuality and thank you for pledging some time to learn with me this term. Do forgive me if this is a bit rough at first, this is the first time that this course is being offered in two different skill levels. But I must say that I'm incredibly excited for this class. You'll take notice of some sweets on all of the tables. They are yours for the taking, have as much or as little as you'd like. As long as my supplies stay plentiful, I'll have something sweet or nice tasting available for you every class."

Timothy himself popped a sweet in his mouth and took a moment to swallow it. He gestured to the board. "This course is going to start with us diving into Critical Thinking Arithmetic Principles. This is the jumping off point for the entire course so in a great many ways, this will be the most important lesson of the term. Critical thinking skills are just as they sound: deep and important experiences of thought and revelation. Critical thinking involves bringing in what you know and applying it to a current situation."

Timothy pointed to one of the bowls of sweets on a table. "For example, these sweets. Let's pretend that I didn't tell you at the start of class that you are free to eat the sweets. You came into the room, sat down, and took notice of them. I said nothing about them. As I walked to the front of the room, I took a sweet for myself but once more did not make any mention of them. The question you would likely be asking yourself is a relatively simple one: 'Can I have a sweet?' That's a very basic question. But it has deeper roots than on the surface. That basic question leads to more complex ones such as 'Is it appropriate for me to eat a sweet in class?' or 'Did the professor leave these here by accident?' Using critical thinking, we can answer all three questions. Let's start with one of the more complicated questions: 'Did the professor leave these here by accident?' You don't know for sure if I did or I did not. But use what you know. You all saw me lean over and take one. That proves that I know that they are there. You also saw me leave them on the desk, I didn't take them away. So you can conclude that I didn't leave them there by accident. Can anyone wager a guess as to how you could answer the question of 'Is it appropriate for me to eat a sweet in class?"

He’d forgotten his book and had to run back upstairs halfway through breakfast to get it, so he was winded by the time he made it to Arithmancy. Even with the extra sprint up to and back from Gryffindor Gene was early, which was the only benefit of needing to plan his actions thirty minutes in advance, and he slouched in early and dropped his book heavily on a desk near the front, flung his bag to the floor and then himself into a chair. There was candy in a bowl in the table; it didn’t have anything he actually wanted to eat but he helped himself to a handful anyway, scattered it on his textbook and picked out the biggest one to eat first.

He was insulted, on principle. They were not children and their respect would not be bought with candy. Not that Gene was going to refuse the candy. (He couldn’t help it, was all. He hadn’t finished breakfast. And he had a biological need to suck on hard candy when given the opportunity.) Only after class started did Winchester actually give them permission to take the candy; Gene, who was on his second piece, surreptitiously brushed the rest into one hand. Hopefully it wouldn’t be brought up again, and hopefully nobody was watching him. How embarrassing.

He’d been hopeful for Advanced Arithmancy to be cool and exciting and fascinating, to finally prove that it was a far better elective than Care of Magical Creatures (take that, Henry) but the first lesson was, evidently, about thinking. Forget giving them candy— this stung. Gene knew how to think. He was seventeen. He’d been thinking for as long as he could remember. He'd been overthinking for as long as he could remember.

(Of course, he hadn’t been thinking about any of the questions Winchester was rattling off as though they were the kind to naturally occur to anybody. He’d taken the candy without permission, and was now starting to doubt that he had been thinking for as long as he remembered.)

But Gene was good at Arithmancy, usually, so Gene was going to suck up his pride and figure out a way to answer Winchester’s question, even though he hadn’t personally put any thought into whether he was allowed to take the candy at all. Probably his uniquely uncritical thinking process was helpful for their purposes, anyway. Gene was a proponent of not coming by an answer with established means. And he was a little bit of a dick, too, so he raised one hand and didn’t wait to be called on before he said, “I mean, I answered it myself by eating one and not being immediately thrown out of class.”

It was the start of a new year and thus a start of a brand new Amelia. She had changed her style of clothing over the summer to reflect her change in appearance due to a new haircut and slight growth over the break. It felt right to enter the halls with a new sense of self, but her fears would of course still ring true in her heart. She was now the eldest Hollingberry at Hogwarts now which meant she had to make sure she was able to keep a strong and confident front much like her siblings prior. Especially since her youngest sister would be entering in the upcoming year so she needed to prove not to just herself, but to Sarah that she wasn't someone who could be pushed around anymore.

Things would be a lot easier without her sister Heather constantly belittling her in the halls and the fights they used to get into. She felt like a weight had been lifted from her shoulders finally and she could blossom as a person finally. So with a jump in her step, she headed to one of her first classes of the year. This was a class she much enjoyed the subject on and was quite decent at overall. Arithmancy used a lot of her brain power in her opinion and back in the day would distract her from the woes she had, but now it could be something she could be passionate about and maybe one day take into a career form.

So she arrived into class in a punctual manner doing her best to make a good impression on her professor and sat at one of the triangular tables close to the professor so she could give him her undivided attention. She placed her book bag on the floor after withdrawing her text and placing it neatly in front of her and that's when she noticed the object in front of her. A bowl of candy was in the center of the table in which she was sitting and a quick scan of the classroom told her each table had one. This was a bit strange to the young Ravenclaw, but she tried her best not to let her thoughts wander.

She was joined by an older student shortly after whom she didn't recognize, but by the color of his robes, he was of the Gryffindor house. She couldn't remember really socializing with many of the lions in the past, at least not family members, and gave him a curt nod when he sat down before facing the professor. She then saw the boy's hand reach for the candy and Amelia arched an eyebrow in intrigue. They hadn't been instructed to take a candy so for all they know they could be a decoration or some sort of task involved in the lesson. She didn't complain though as she noticed he didn't get in trouble and he seemed fine after taking a piece so it wasn't cursed or anything. Contemplating on whether she should take on herself Amelia decided it would be best to ignore it for now. Though if she could she might take one after class for a light snack between this and her next one.

Then class began and the professor began his speech addressing the candy in the room. So they were allowed to take one and Amelia gave a quick glance back over to her tablemate and felt relief that she hadn't sat next to a possible troublemaker. She was right though in thinking that they would be apart of the lesson though which made her a bit giddy. So the lesson would be on the thinking of situations and how to interpret them without proper instruction or prior knowledge. This would be a data collection lesson then and to infer hidden meanings behind actions and situations which was quite an interesting concept to start the school year with. Then again they were in the advanced class of Arithmancy so Amelia wasn't that surprised by the notion of it.

When the professor started asking questions this put Amelia in a trance as she ran the questions through her own head trying to figure out the many ways to interpret the situations. She didn't get much time to think through as her tablemate began with his own statement about the situation they were in. She had to agree with him, though not of his methods, that he didn't, in fact, get in trouble for. She decided she would contribute to the conversation as she raised her own turn and began speaking soon after. "There is also the fact you left a dish at each table. If this was an accident then there wouldn't be one for each table and instead, most likely be one at a single table. As for the appropriate levels of us eating the sweets in class, they wouldn't be out in the first place if they weren't in fact meant to be eaten wouldn't they?" She paused for a second before continuing, "In fact leaving them there would infer that the lesson would involve the objects in question. If they weren't meant to be eaten you would have passed them out once the lesson started or after you explained the lesson at hand. So with this, we could determine they were there for us to consume if we so wish to or not." Amelia stopped talking and realized she said quite a bit more than she was used to in class and gave off a slight blush before quietly asking, "is this correct?"

Almost as soon as his question left his mouth, a Gryffindor, a student named @Gene Horowitz, answered with sarcasm. Though in truth he should have been angry, Timothy laughed. "Well, yes you do have a point." Timothy honestly found the remark humorous. Many professors would have handed out a detention slip or issued some sort of other punishment. But to Timothy, that wasn't necessary. After all, these were teenagers. That's just how teenagers speak, that's just how they act. Timothy was guilty of doing the same thing when he was teenager. In a total opposite response, a Ravenclaw named @Amelia Hollingberry gave just the answer Timothy was looking for. "Miss Hollingberry, that's absolutely correct. You're making our house very proud. That was exactly what I was looking for. I think 5 points for Ravenclaw would compliment the work very nicely."

Timothy moved to the front of the room. "What Miss Hollingberry did in her rationale was observe the situation and circumstances surrounding her. Now, this example of the sweets on the table is a simple one. It's uncomplicated. But the same exact method that was worked through by myself and her and I'm sure in many of your own minds can be applied to incredibly intricate and complicated equations and questions when working with Arithmancy. Arithmancy is a science that isn't easily defined. Some see it as a branch of Divination, some see it as a glorified practice of numerology. The history of Arithmancy is complicated and there are several decades worth of information gaps. In the dark ages, Divination and Arithmancy were not viewed upon fondly. They were viewed then as we view dark arts today. The creation of those gaps in knowledge concerning the subject directly led to the two principals of Arithmancy that we know today: Exaction and Estimation. Estimation is the only option for certain corners of Arithmancy. However, Exaction is applicable to all of Arithmancy. In our normal, every day lives, we tend to use Exaction far more often than Estimation. But in the case of advanced Arithmancy, we're more concerned with Estimation. In order to understand Estimation, we must understand the process in which we think about what answer we're looking for. Now that we've gotten the warm up out of the sweet dishes out of the way, I'd like to get into an activity working with a bit more complicated components. Please, stand up from your seats, move the seats to one of the walls in the room, and stand at stand by them.."

Timothy waited for the students to comply. He erased the lesson title on the blackboard and replaced it by drawing three columns. One read Given while the second read Gathered and the final read Impacting Fluctuation Barriers. "This is an example of what is known as a McGregor's Table. It was developed in the sixteenth century by an Artithmist named Bartty McGregor. He is most well known for this system, one that organizes the information and thoughts in order to complete any range of equations or exercises." He pulled his wand from off his desk and with a simple swish, all of the desks stacked, one on top of the other, in the back left corner of the room leaving the the remaining floor space in the room bare. "This exercise that we're going to do is a Diply Procedure. Diply Procedures are a series of tests that establish proofs for Arithmetic concepts. This particular test is used as proofs to Estimations. What we're going to do is this. I am going to supply you with three definitive pieces of information. Just those three. Those pieces of information are this: 1. This classroom is 625 square meters large.2. The length of the room is 25 meters and the width is 25 meters.3. The average person's stride, which is the length of one step, is 762 millimeters.

The question we are presented with is this: How long do you think it will take for you to walk from one side of the classroom to another? Now, you'll notice this third column which speaks of impacting fluctuation barriers. This is where critical thinking comes into play. In this column, we need to write down what outside factors will cause the answer to our question to be different from person to person. This means we have to think about what we know. Can anyone tell me what one major thing will cause almost all answers to the main question be different depending on the person?"

Obviously if Gene had wanted to be taken seriously, he would have answered the question seriously instead of just rattling off the first thing he’d thought to say— but it smarted all the same when Winchester laughed. It wasn’t like Gene had meant it as a joke— he’d been right, hadn’t he? You could be right and funny. But whatever— it didn’t matter. (But Amelia Hollingberry was a show-off.)

Most of what Winchester went on to say was crap they’d been over in the four years Gene had already taken Arithmancy, so he felt safe zoning out completely until he noted that his classmates were getting up and moving their chairs. Rearranging the classroom wasn’t enough, it seemed— they all had to rearrange it further. They’d probably have to rearrange it back to normal, too. Grudgingly Gene resigned himself to the miserable necessity of paying attention to the rest of the lesson, slouching at the side of the classroom. He inspected his hands for an easy target and then bit carefully at the side of one knuckle.

Critical thinking was something that came naturally to him, as a person who criticised things constantly. He was insecure, and he was whiny, and had seventeen years’ experience under his belt. Obviously this wasn’t necessarily the same level of critical thinking that Winchester wanted them all to practice, but that wasn’t too difficult either, with some mental math. 762mm was around three-quarters of a metre, it took four steps to walk three metres, and twenty-four was three times eight, and four steps per three metres was four times eight, and so it would take around thirty-two steps to walk across the room. Give or take a few.

And that was Estimation, not even close to Exaction, but Gene wasn't about to do that on paper, so he didn't care.

And one major variable in that was a question Gene had been born ready to answer; he raised his hand again, and said “People will have shorter or longer strides, probably based on height.” (And Gene was taller than Winchester, which had happened sometime last year and given him more satisfaction than he liked to admit.)

sorry to post again so soon he just really cares about not coming off as a dumbass in class

Amelia couldn't help but turn beet red at the professor's praise. It was nice to be given positive affirmation to her answers. Sure they were other professors who were kind and congratulated Amelia on answering a question right, but the fact she wasn't attacked about it by her older sister anymore was a very nice change of pace. She lowered her head her shyness showing a bit in response to the professor, but a small smile tinged her face as her hair fell past her eyes. She took a deep breath as the professor continued speaking before raising her head again.

The talk about Estimation and Exaction was certainly a fascinating subject matter to the Ravenclaw girl. Though she was more focused on Exaction herself when she studied her subjects so when he said they would focus on estimation she felt a tinge of fear. She hated estimating it was never exact and sometimes concrete evidence would differ from estimated evidence. She may have been able to think about estimations at times, but she preferred to make them exactions. Amelia felt this lesson might end up being a bit difficult for her today, but she focused her mind as much as she could and listened to the professor continue to speak.

They were doing something physical with their lesson? This didn't bode well with Amelia as she wasn't that strong. Not risking showing her true weakness though she took out her wand and moved her seat to a nearby wall. She stuck close to the chair and tried to rearrange herself so it looked like she was lifting it herself in hopes she wasn't breaking any rules by performing magic to do this simple task. Amelia hoped she had done this well enough to keep it hidden as she lowered the chair down and hid her wand in her robes as quickly as possible.

Professor Winchester then drew three columns on the blackboard that filled up with more information on the practical lesson they might be attempting. Some math would be involved in this experiment of a lesson and Amelia became calmer than she was when he told them to get up from their seats. Math was something she did well at as it had exact numbers and definitive answers. Wait weren't they focusing on estimation though? Why were they given numbers and figures when discussing estimations? Couldn't they just calculate and get an exact number instead? That would be better right? Amelia felt a bit confused about what the professor was asking until he addressed the third part of the column.

So they had to figure out what the fluctuating barriers were. This got Amelia's brainstorming as this could be anything and did explain why the information they were given earlier. This was a trick question, wasn't it? They were given more information then what was needed for the question that was proposed. The first two were obvious for figuring out as they were constant numbers that wouldn't change thus giving them an exaction, but when Amelia looked at the third one this had more possibility of being an estimation.

Amelia was just about to answer the professor's question when the lion from earlier answered first again. He brought up the situation of people's strides which was a fair answer but in her mind an incomplete one. So she quickly tacked on to the end of his response, "There is also the plausible factors of where people start in the room in relation to others. If two people start at opposite sides of the room and go in a straight line they may end up bumping into each other which will cause one or both of them to stray off course making their answer differ even more than just the height issue." Amelia took a quick breath as she looked around the room at the other students to gather more information on the situation at hand. "We also have to take into consideration fluctuating speeds in which people walk or pace themselves," Amelia said looking back at the professor and continued, "Though we may all walk we may stumble upon our own feet or something catches our eye forcing us to slow down or even speed up if the drive of competition rages in someone's mind." She looked at the lion who answered earlier and gave him a bashful smile before adding, "finally we also have to take into consideration outside factors such as Peeves coming into the classroom and causing everyone to go into a panic." Amelia shivered at this thought and quickly shut herself up and couldn't finish what else she wanted to say on the subject at hand.

Timothy was taken aback but still amused by @Gene Horowitz 's answer. He was a little more mature about this time, but Timothy saw satisfaction. And truthfully, he found it rather funny. "Yes, Mr. Horrowitz, yes indeed." Almost immediately after the words left his mouth, @Amelia Hollingberry spoke up and gave another fantastic answer. "Correct again Miss Hollingberry. You're having a fantastic day today I see." Timothy smiled. "So what Mr. Horowitz and Miss Hollingberry are saying is quite right. Most of you all are different heights, have different stride lengths. In Arithmancy, we call those factors Impacting Fluctuation Barriers. There is no standard equation or formula for these barriers and the reason for that is simply because Arithmists throughout history simply cannot agree how to appropriately work them into generalized equations or formulas. It is because of that fact that critical thinking becomes imperative."

Timothy walked to the center of the room and took his wand out. He counted the number of heads in the room and when he had his number, he gave a simple flick of his wrist. On command, small slate boards complete with a piece of chalk flew out of a cabinet by the door and settled in a neat line on the opposite side of the room from the students. Timothy did a double check on the count to make sure. Indeed, there was one slate and one chalk per student. Timothy gave another flick of his wand and a line appeared on the floor right at the toes of the students.

"Now that you know the basics of critical thinking as well as the basics of the concept of Impacting Fluctuation Barriers, you are going to put that knowledge to work using this Diply Procedure. When I say go, you will step over this line at your toes and walk across the classroom to one of the slate boards. On the way over, take notice of your surroundings. Take notice of if there are obstacles in your way, if you have to change your intended destination to another board because someone got the one you were going to first, things like that. When you get to a board, draw a McGregor's Table and fill in Given and Impacting Fluctuation Barriers. This particular version of the exercise is going to be using Estimation, so in the gathered write down your estimations of number of steps, your potential stride lengths, information such as that. Once that is done, attempt to use all of that information to answer the question of how long did it take you to get from one side of the classroom to the other. You're advanced students so I trust that you'll be able to form your answers on your own. Once you've got all of your items written down, raise your hand and I will come by and check."

This was the thing: Winchester’s three mathematical facts had been clear-cut and exact, had implied a simple calculation. Length of room, area of room (and that had been stupid— they were all old enough now to square twenty-five and they weren’t using that in their calculations), and average length of stride. It was a class about math and Gene was annoyed that they were all acting as though these ostentatious answers about Peeves or tables in the middle of the room or the drive of competition raging in their minds were at all relevant.

It would have been entirely reasonable to assume that they were old enough not to immediately panic at the sight of Peeves like Amelia Hollingberry was suggesting they would, and smart enough not to walk into each other. And walking faster or slower didn’t affect the number of paces, this wasn’t about their speed. It rankled that Winchester was acting as though these nonmathematical obstacles were things they should naturally think about in a math class. Gene considered himself pretty good at Arithmancy, usually: that Amelia and Winchester were both throwing information he had never thought to consider was driving him a little mad. With the three pieces of given information, his answer had been absolutely reasonable, not incomplete.

Anyway, no doubt this was part of Winchester’s diabolical lesson plan, to show that nothing had an answer, but he couldn’t help finding it funny that the instructions were vague enough that Gene could walk the perimeter of the class to get the slate without being out of line. As they were set loose to their calculations he hesitated for a bit, torn between testing his thirty-two steps hypothesis and being contrary.

It was annoying him that Winchester had phrased it as “how long will it take?” rather than “how many steps?,” but at heart he wanted to prove that he was a good student, so Gene set off in a nice gangling walk to the other side of the classroom, then stooped to pick up chalk and a slate (what year was it, 1910?) and slapped the slate against the wall to scrawl his data:

He gave this a skeptical once-over. It was barely legible, but that would be fine, Gene could read it to Winchester. And Winchester had asked them to estimate the number of paces, but Gene could count to thirty, so that was unnecessary. And Winchester had also asked how long it had taken, but they hadn’t been given timers or been told to look at the clock or anything, and time and number of paces were completely different concepts anyway, so that was unnecessary too. This had to be a reasonable answer; he raised his hand.